Dynamics of astrophysical discs

Lecture 4

FDEPS, Kyoto

01.12.11

Gordon Ogilvie

gio10@cam.ac.uk

DAMTP, University of Cambridge
2011年12月2日金曜日

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2D compressible dynamics in shearing sheet● 2D compressible sheet: inviscid, self-gravitating● Surface density Σ(x, y, t)● 2D pressure P (x, y, t)● Relate to vertically integrated quantities

�

ρ dz,

�p dz

but only a model, not derivable exactly from 3D equations● Basic equations:∂Σ2
Φ = −ΩSx+ ∇ · (Σu) = 0∂t1∂u+ u · ∇u + 2Ω × u = −∇Φ − ∇Φd,m − ∇P∂tΣ
2∇Φd = 4πGΣ δ(z)● Disc potential Φd (x, y, z, t) satisfies● Then evaluate in midplane: Φd,m (x, y, t) = Φd (x, y, 0, t)● Assume barotropic relation P = P (Σ) for simplicity
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2D compressible dynamics in shearing sheet● Solve Poisson’s equation in Fourier domain:� ∞� ∞
Σ(x, y, t) dx dy etc.
Σ̃(kx , ky , t) =−∞−∞∇2 Φd = 4πGΣ δ(z)��2∂2⇒ −k + 2 Φ̃d = 4πGΣ̃ δ(z)∂z2πGΣ̃ −k|z|e⇒ Φ̃d = −k2πGΣ̃⇒ Φ̃d,m = −k(k �= 0)

k = (kx2 + ky2 )1/2

so that�∂ Φ̃d∂z�0+0−● k = 0 component gives no horizontal force anyway

2011年12月2日金曜日= 4πGΣ̃2D compressible dynamics in shearing sheet

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● Conservation of potential vorticity / “vortensity”:
1∂u+ u · ∇u + 2Ω × u = −∇Φ − ∇Φd,m − ∇P∂tΣ● Use identity (∇ × u) × u = u · ∇u − ∇( 12 |u|2 ) :∂u+ [(2Ω + ∇ × u) × u] = ∇(· · · )∂tsince P = P (Σ)∂(∇ × u) + ∇ × [(2Ω + ∇ × u) × u] = 0∂t● Use identity ∇ × (A × B) = B · ∇A − A · ∇B + A(∇ · B) − B(∇ · A) :��∂+ u · ∇ (2Ω + ∇ × u) = (2Ω + ∇ × u)(∇ · u) since 2D∂t��
1 ∂+u·∇ Σ= −(2Ω + ∇ × u)
Σ ∂tDq2Ω + (∇ × u)z=0⇒where q =DtΣ2011年12月2日金曜日

2D compressible dynamics in shearing sheet

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● Conservation of potential vorticity / “vortensity”:
Dq=0Dt2Ω + (∇ × u)zwhere q =Σ● Recall u = −Sx ey + v :��∂∂− Sx+v·∇ q =0∂t∂y2Ω − S + (∇ × v)zq=Σ● Unlike incompressible 2D case, vortex dynamics not the whole story● Vortical disturbances are coupled to acoustic ones

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2D compressible dynamics in shearing sheet

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● Linear stability of uniform 2D self-gravitating sheet∂ΣΦ = −ΩSx2+ ∇ · (Σu) = 0∂t1∂u+ u · ∇u + 2Ω × u = −∇Φ − ∇Φd,m − ∇P∂tΣ
∇2 Φd = 4πGΣ δ(z)● Basic state: Σ = cst,u = −Sx ey● Linearized equations for perturbations Σ� , v, etc.:��∂∂�2 �− SxΣ� + Σ∇ · v = 0
P = vs Σ∂t
∂y�
�∂
1
∂
�
− Sxv − Svx ey + 2Ω × v = −∇Φd,m − ∇P �∂t∂yΣ∇2 Φ�d = 4πGΣ� δ(z)2011年12月2日金曜日

sound speed vs2D compressible dynamics in shearing sheet● Solutions are shearing waves:��Σ� (x, t) = Re Σ̃� (t) exp[ik(t) · x]etc.

● Amplitude equations:dΣ̃�+ Σ ik · ṽ = 0dt
��
�dṽx
Σ̃
− 2Ωṽy = −ikx Φ̃�d,m + vs2dt
Σ�
�
�dṽy
Σ̃
+ (2Ω − S)ṽx = −iky Φ̃�d,m + vs2dt
Σ�
2πG
Σ̃
Φ̃�d,m = −k�(2Ω−S)
Σ̃ikṽ
−ikṽ
x yy x�−● Vortensity perturbation q̃ =ΣΣ2dq̃ �= 0 as expected [exercise]satisfiesdt

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2D compressible dynamics in shearing sheet

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● Consider axisymmetric waves: ky = 0, kx = cst, k = |kx |● Amplitudes ∝ e−iωt−iω Σ̃� + Σ ikx ṽx = 0−iωṽx − 2Ωṽy = −ikx

�2πGΣ
2vs −|kx |−iωṽy + (2Ω − S)ṽx = 0�Σ̃�Σ● Multiply second equation by iω and eliminate Σ̃� and ṽy :��
2πGΣ
222ṽxω ṽx − 2Ω(2Ω − S)ṽx = kx vs −|kx |● Deduce dispersion relation for “density waves”:
ω 2 = κ2 − 2πGΣ|kx | + vs2 kx2● Also vortical solution ω = 0, ṽx = 0 : zonal flow / geostrophic flow
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2D compressible dynamics in shearing sheet

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● Dispersion relation for density waves:
ω 2 = κ2 − 2πGΣ|kx | + vs2 kx2inertial

acoustic

self-gravity(restoring forces)(destabilizing)● “Acoustic-inertial waves”● Disc is unstable to axisymmetric disturbances if ω 2 < 0 for some kx● ω 2 is minimized with respect to |kx | whenπGΣ
⇒ |kx | = 2vs��2(πGΣ)
1222(ω )min = κ −=κ 1− 22vsQvs κ● Gravitational instability if Q < 1 , where Q =πGΣ(Toomre stability parameter)
0 = −2πGΣ + 2vs2 |kx |2011年12月2日金曜日

2D compressible dynamics in shearing sheetvs κ● Gravitational instability if Q < 1 , where Q =πGΣ
● Toomre stability parameter Q :● An inverse measure of self-gravity● A measure of temperaturevs κ
Q=πGΣ

2011年12月2日金曜日

product of stabilizing effects(short and long scales)
destabilizing effect

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2D compressible dynamics in shearing sheet
ω̂κ

Q=∞
Q=2

2

Q=1

1

�2

�1

1vskκ
2

�1
�2vs κ
Q=πGΣ
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2D compressible dynamics in shearing sheet
Nakagawa & Sekiya (1992)
Non-vortical perturbations
Oscillating at ends of swing
Over-reflection due to wave action conservation|I|2 = |R|2 − |T |2

positive wave action
negative wave action2011年12月2日金曜日

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2D compressible dynamics in shearing sheet
Maximum transmission coefficient
Swing amplification due (mainly) to self-gravity

Nakagawa & Sekiya (1992)

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● Occurrence of gravitational instability:● If Q < 1 , disc tends to form rings(axisymmetric instability, exponential growth)● If 1 < Q � 1.5 , disc tends to form spiral waves or clumps(non-axisymmetric instability, transient growth)● Since Q ∝ vs ∝ T 1/2 , thermostatic regulation is possible:
instability → motion → dissipation (shock/viscous) → heating● Two possible outcomes of gravitational instability:● Fragmentation: formation of gravitationally bound objects(clumps...moonlets / planets / stars)● Gravitational turbulence: sustained activity of non-axisymmetricdensity waves (e.g. “self-gravity wakes” in Saturn’s rings)● Efficient cooling promotes fragmentation, or enhances the efficiencyof gravitational turbulence, since cooling balances viscous heating
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Gravitational instability
Nonlinear outcome in razor-thin disc with heating and cooling(Gammie 2001)
energy2011年12月2日金曜日

angular momentum transport

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Gravitational instability
Gravitoturbulent state (slow cooling)

Gammie (2001)2011年12月2日金曜日

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Gravitational instability
Fragmentation (rapid cooling)

Gammie (2001)2011年12月2日金曜日

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Gravitational instability● 2D nonlinear simulation with cooling time β/Ω (S.-J. Paardekooper)β=4

β = 10surfacedensity
ΣsoundspeedvsToomre
parameterQ

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Gravitational instability● 2D global simulation with cooling (S.-J. Paardekooper)β = 102011年12月2日金曜日

Gravitational instability

4-19
FDEPS, Kyoto© GIO 2011● 3D simulations of Saturn’s rings (inelastically colliding particles)(Salo 1992)

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Satellite–disc interaction

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● Common problem:● Orbiting companion, e.g. on circular orbit within disc
● Gravitational (rather than hydrodynamic) interaction with disc
● Perturbs orbital motion and excites waves● Calculate exchanges of energy and angular momentum● Determine orbital evolution of satellite (migration, etc.)

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Satellite–disc interaction

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Satellite–disc interaction

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● Test particle dynamics in xy plane, in local approximation(fluid dynamics more difficult, but results are similar in some ways)∂Ψẍ − 2Ωẏ = 2ΩSx −∂x∂Ψÿ + 2Ωẋ = −∂y● Satellite on circular orbit at reference radius (xs = ys = 0) :
Ψ = −GMs (x2 + y 2 )−1/2

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Satellite–disc interaction∂Ψẍ − 2Ωẏ = 2ΩSx −∂x∂Ψÿ + 2Ωẋ = −∂y● General solution in absence of satellite potential:...
x = −4Ω2 ẋ + 2ΩS ẋ = −κ2 ẋ
� −iκt �⇒ x = x0 + Ar cos κt + Ai sin κt = x0 + Re A e
�
�2Ω
−iκt
Re iA ey = y0 − Sx0 t −κ● Guiding centre (x0 , y0 − Sx0 t)● Complex epicyclic amplitude A = Ar + iAi

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Satellite–disc interaction● To express “orbital elements” in terms of position and velocity:� −iκt �x = x0 + Re A e�� −iκt ��−iκt= κ Im A eẋ = Re −iκA e� −iκt �2ẍ = −κ Re A e⇒ Ae−iκt

�ẍ
iẋ
=− 2 +κκ�

2Ωiẋ iκte
⇒ A = − 2 (ẏ + Sx) +κκẍ2Ω2Ω
x0 = x + 2 = x + 2 (ẏ + Sx) = 2 (ẏ + 2Ωx)κκκ

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Satellite–disc interaction● Canonical y momentum (per unit mass):
κ2x0 = cst
py = ẏ + 2Ωx =2Ω● Energy (per unit mass):
1 2ε = (ẋ + ẏ 2 ) − ΩSx2224Ω● Use κ2 |A|2 = ẋ2 + 2 (ẏ + Sx)2 :κ

1 2 2 2Ω2
1 22ε = κ |A| − 2 (ẏ + Sx) + ẏ − ΩSx22
κ2
1 2 2 ΩS= κ |A| − 2 (ẏ + 2Ωx)22κ
1 2 2 ΩS 2= κ |A| − 2 py = cst2κ

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Satellite–disc interaction● In the presence of a satellite potential:∂Ψṗy = −∂yε + Ψ = cst
��iẍ 2iΩ2Ω−(ẏ + Sx) − ẋ eiκtȦ = − 2 (ÿ + S ẋ) +κκκ��2Ω
i= − 2 (ÿ + 2Ωẋ) + (ẍ − 2Ωẏ − 2ΩSx) eiκtκκ��i ∂Ψ iκt2Ω ∂Ψ−e=2κ ∂yκ ∂x

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Satellite–disc interaction● Consider the unperturbed “circular” orbit ( A = 0 )x = x0 = cst

y = −Sx0 t● Calculate ∆A in linear approximation:��i ∂Ψ iκt2Ω ∂Ψ22 −1/2Ψ=−GM(x+y)−eȦ =s2κ ∂yκ ∂x�
�ix iκt2Ωy22 −3/2e= GMs (x + y )−2κκ
�
�
2Ω
GMs
2 2 −3/2iκt
1−iSte≈ −i(1+St)
κx20
κ� ∞
∆A =Ȧ dt−∞

GMs= −iκx20
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� ∞−∞(1 + S 2 t2 )−3/2��
2Ωcos κt +St sin κt dt
κ4-28FDEPS, Kyoto© GIO 2011

Satellite–disc interaction��2Ω
cos κt +St sin κt dtκ� ∞GMs
2 2 −3/2∆A = −i(1+St )2
κx0 −∞
� ∞(1 + x2 )−3/2 cos kx dx = 2kK1 (k)● Let f (k) =−∞(k > 0)modified Bessel function● ThenGMs∆A = −iC
κSx20C=f● For Keplerian orbits ( κ/S = 2/3 ),

�κ�
S�κ�

2Ω �−fκSC ≈ 3.359● So encounter with satellite excites an epicyclic oscillation
at first order

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● Long before and after the encounter, Ψ → 0● Since ε + Ψ is exactly conserved, ∆ε = 0 in the encounter4
1 2 2 ΩS 2κ∆(|A|2 )● But ε = κ |A| − 2 py , so ∆(p2y ) =2
κ2ΩS● Assume a “circular” orbit before the encounter:κ2x0
A = 0, py =2Ω● Then, after the encounter:��244GMs
κ 2GMsκ
2C, py ≈A ≈ −iCx0 +22
κSx04Ω
2ΩS
κSx20{κ2(CGMs )2x0 +⇒ py ≈2Ω
2S 3 x50∆py correct to second order● Irreversibility / dissipation implicit in assuming circular initial orbit
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Satellite–disc interaction
y40

20

0.8!20!40

!60

!80

!100

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1.0

1.2

1.4x

NASA
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NASA
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Satellite–disc interaction● Simplified version: “impulse approximation”

Sx0

GMs 1∆v⊥ ≈x20 S2∆(v⊥) + ∆ (v�2 ) = 0Ms�

GMs
Sx20

�2+ 2Sx0 ∆v� ≈ 0(GMs )2∆v� ≈ −2S 3 x50x0

2011年12月2日金曜日(energy)(lacks C 2 factor)

Satellite–disc interaction● y force on disc per unit x at location x :(CGMs )2=
Σ |Sx|
352S xencounter ratesurface density∝ x−4 sgn(x)● Torque per unit radius is the same ×r0● Satellite experiences an equal an opposite torque● Effect is of second order in Ms● Similar result for density waves (response of a fluid disc)● x−4 divergence is moderated within |x| � H (or Hill radius)

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Satellite–disc interaction● Gravitational interaction is “repulsive”!

transfer of py (or angular momentum)● One-sided torque leads to gap opening if Ms large enoughand ν small enough● Asymmetry leads to net torque on satellite and to migration(usually inwards)
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Satellite–disc interaction

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● Now include periodic nature of y coordinate (Ly = 2πr0 ) :��i ∂Ψ iκt2Ω ∂Ψ−eȦ =2κ ∂yκ ∂x
2πr0T =iκt
S|x0 |= F (t) e∞
�S|x0 |2π−inωt iκt=fn ee=ω=T
r0n=−∞● Add damping of epicyclic motion:∞�
fn e−inωt eiκt − γAȦ =n=−∞● Long-term response:∞�
ifn e−inωt eiκt
A=(nω−κ)+iγ
n=−∞

1κ
x=
● “Lindblad resonances” where, resolved by damping
r0nS
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Satellite–disc interaction

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● Lindblad resonances:
r0nS● In a Keplerian disc, LRs correspond to orbital commensurabilitiesnΩ=Ω0n−1● In a fluid disc, density waves are launched there(wave emission resolves singularity in response)
2011年12月2日金曜日

Cuzzi et al.(2010)

Cuzzi et al.(2010)

Cuzzi et al.(2010)2011年12月2日金曜日

reference1
・NASA
・Nakagawa, Y. & Sekiya, M,1992:Wave action
conservation, over-reflection and over-transmissionof non-axisymmetric waves in differentially rotating
thin discs with self-gravity, MNRAS, 256, 685-694
・Gammie, C. F. ,2001:Nonlinear Outcome ofGravitational Instability in Cooling, Gaseous Disks,ApJ, 553, 174-183
・Paardekooper, S.-J. , unpublishedreference2
・Salo 1992:Gravitational wakes in Saturn's rings,Nature, 359, 619-621, supplimental material・Cuzzi, J. N. , Burns, J. A. , Charnoz, S. , Clark, R. N. ,Colwell, J. E. , Dones, L. , Esposito, L. W. ,Filacchione, G. , French, R. G. , Hedman, M. M. ,Kempf, S. , Marouf, E. A. , Murray, C. D. , Nicholson,
P. D. , Porco, C. C. , Schmidt, J. , Showalter, M. R. ,Spilker, L. J. , Spitale, J. N. , Srama, R. , Sremčević,M. , Tiscareno, M. S. and Weiss, J. , 2010: An
Evolving View of Saturn’s Dynamic Rings, Science,327, 1470-1475